Umpire's Call

Is This the Solution to the Duckworth Lewis problem?

If you follow cricket then you have surely heard of the Duckworth Lewis method.  It’s a clever statistical method that for nearly twenty years has been used to adjust the run target of the team batting second when a match has been interrupted by a long rain break.   A team that originally needed to score another 150 runs in 25 overs to win a match might, because time has been lost to rain, be set a revised target of 80 runs in ten overs.   Many experts in the game regard it as far superior to any other officially adopted system that has ever been used to cope with a rain-affected game.  It even earned its two inventors MBEs in the Queen’s Birthday Honours in 2010.

Yet many cricket fans are still not happy with it.  That’s because sometimes the D/L calculation seems to favour one of the teams.  In June 2015, England were on their way to what looked a likely victory against New Zealand at The Oval. Then it rained.  Suddenly a target of 54 off 39 balls was adjusted to 34 off 13 balls.  England lost the game, and the England captain Eoin Morgan was clearly not happy.  There were calls for the system to be changed.

But there’s a bigger reason why the Duckworth Lewis system is unpopular, and it’s nothing to do with how technically good the D/L method is.  The real problem is that the D/L method has disempowered most cricket fans.  In days of old, a fan could tell you at any point how they reckoned the match would unfold. Now when it rains the players and fans are at the mercy of a computer, a black box that tells them what happens next, or even who has won the match, through a process they struggle to understand and which they aren’t able to challenge.  Cricket has lost the human touch.

It doesn’t matter what rival mathematical system is invented that is ‘better than’ Duckworth Lewis, it will still be a black box.

So I’d like to propose a different solution, that returns cricket into the hands of the most important people: the players and the umpires.  (I’d like to thank my colleagues Dr Hannah Fry and Matt Parker who came up with a couple of vital insights when we discussed this idea.)

The basis of our idea is simple.  When cricketers look at the Duckworth Lewis calculations and say “this isn’t fair” (as Eoin Morgan did, for example), it surely means that those cricketers have an idea of what is fair.  If 34 runs off 13 balls wasn’t fair, what did Eoin Morgan think was fair? 28 runs maybe? He must have had a view.

If you were to ask the captains of the two teams they would almost certainly disagree about what was fair.  The team that’s batting will tend to think a lower target is fair, the bowling team will think a higher target is fair.  It’s a bit like two kids cutting a cake and both wanting the bigger bit.  What we need is something equivalent to the “I cut, you choose” strategy that resolves arguments about cake sharing.  Who can decide between the two captains?  Surely it should be the people who make decisions in all other aspects of the game of cricket: the umpires.

So here is the idea we came up with.

When there’s a rain interruption, the umpires work out how many overs the time remaining allows (as already happens).  They then privately decide on a run target that is fair to both teams.  To help them, they are given the Duckworth Lewis calculation, but they are not bound by it.  If they think it’s too generous to one team or the other, they can adjust it.

Meanwhile, the two captains also decide on what THEY think is a fair run target.  The captains each write down their target on a piece of paper and hand it to the umpires.  And here’s the twist: whichever captain’s target is closest to what the umpires have decided is fair gets their wish.

So let’s suppose that the umpires have set a revised target of 200 runs.  The batting captain thinks 190 is fair, the bowling captain thinks 206 is fair.  Since the bowling captain is closer, he (or she) gets their wish, and 206 is the agreed target.

You might think this seems very unfair to the team that loses in the bidding.  But it’s only unfair because the batting team was far too greedy in trying to lower the target.  What they should have done is get as close as possible to what they thought the umpires would think was fair.  And since the role of the umpires is to be completely impartial, it should always be in the interests of both captains to get as close as possible to the umpires’ figure.

We’ve debated whether giving the ‘closer’ captain their wish in full is too extreme.  The ‘losing’ captain might disagree with the judgment of the umpires, and feel this has been compounded by going towards the other captain’s figure.  If this is a problem (and we don’t think it should be) the reward to the ‘closer’ captain could be damped down.  Instead of going all the way to the closer captain’s figure, maybe the target would go half way to their figure (so in the above example, the target after rain would be adjusted to 203 instead of 206).  Or maybe the target should only ever move by one run in the direction of the closer captain, so it would be 201 instead of the umpires’ 200.  In the unlikely event that the umpires’ target isn’t between the two captains’ targets (eg the umpires say 200, and the captains say 190 and 195) then the target is simply set at the figure of whichever captain was closer to the umpires (which would be 195 in this case).

In practice, what would happen is that the adjustment to the target would nearly always be close to the Duckworth Lewis figure.  Only in situations where Duckworth Lewis was perceived by the umpires to have got it wrong would the target move away from the D/L value, and this would be a good thing.

But there would be a more important effect, too.  Cricket fans and players would now know that the agreed target in the game was no longer being forced on them by a black box.  Instead, that black box could be over-ridden by human judgment – just as happens in third umpire reviews, where something called “umpire’s call” means the decision of Hawkeye on lbw decisions gives the umpires a little bit of slack.  Fans would know that the decided figure was somewhere between what the batting and bowling captains had thought was fair. Power would return to the players and the umpires. And that’s what cricketers want.